CELE Reinforced & Prestressed Concrete — Reinforced Concrete Beams: FlexureConcept Map

Concept

Singly Reinforced Rectangular Beams

Sub Concepts

• Equivalent stress block (Whitney block)
• Depth of stress block (a)
• Nominal moment capacity (Mn)
• Design strength (φMn)
• Tension-controlled criterion
• Net tensile strain (εt)

Relationship To Central

Foundational design method; most common beam type in practice

Concept

Steel Ratio Limits & Ductility

Sub Concepts

• Balanced steel ratio (ρb)
• Maximum steel ratio (ρmax)
• Minimum steel ratio (ρmin)
• Tension-controlled limit (εt ≥ 0.005)
• Balanced failure condition
• Over-reinforced vs under-reinforced behavior

Relationship To Central

Critical constraints ensuring ductile, predictable failure

Concept

Flexural Analysis (Given Section, Find Capacity)

Sub Concepts

• Compute stress block depth (a)
• Calculate nominal moment (Mn)
• Verify tension-controlled condition
• Apply strength reduction factor (φ)
• Check against applied factored load (Mu)

Relationship To Central

Determines φMn for existing or proposed sections

Concept

Flexural Design (Given Load, Find Steel)

Sub Concepts

• Coefficient of resistance (Rn)
• Steel ratio from quadratic formula
• Required steel area (As)
• Bar selection and spacing
• Verify ρmin ≤ ρ ≤ ρmax

Relationship To Central

Inverse of analysis; sizing reinforcement

Concept

Doubly Reinforced Beams

Sub Concepts

• Compression steel (A's)
• Superposition method
• Check if compression steel yields
• Strain compatibility
• When to use doubly reinforced

Relationship To Central

Extension for large moments or deflection control

Concept

T-Beams & Flanged Sections

Sub Concepts

• Flange thickness (tf)
• Effective flange width (bf)
• Stress block location
• Rectangular vs T-beam analysis
• Web width (bw)

Relationship To Central

Common in monolithic slab-beam construction

Concept

Material Properties & Assumptions

Sub Concepts

• Concrete compressive strength (f'c)
• Steel yield strength (fy)
• Stress block equivalent depth factor (β₁)
• Strain limit at ultimate (0.003)
• ACI/NSCP 2015 material specifications

Relationship To Central

Foundation for all calculations

Concept

Failure Modes & Behavior

Sub Concepts

• Tension-controlled failure (steel yields first)
• Compression-controlled failure (concrete crushes first)
• Transition zone behavior
• Ductility implications
• Safety under different loading

Relationship To Central

Explains why design limits exist

To

Steel Ratio Limits & Ductility

From

Singly Reinforced Rectangular Beams

Strength

strong

Relationship

Steel ratios determine whether the beam is over- or under-reinforced, directly controlling failure mode and strength reduction factor φ

To

Singly Reinforced Rectangular Beams

From

Flexural Analysis

Strength

strong

Relationship

Analysis applies the stress block principle and moment equations to compute capacity from known section

To

Flexural Analysis

From

Flexural Design

Strength

strong

Relationship

Design produces section dimensions and reinforcement; analysis verifies that design meets the demand

To

Steel Ratio Limits & Ductility

From

Flexural Design

Strength

strong

Relationship

Design process enforces ρmin and ρmax constraints to ensure ductility and minimum cracking control

To

Singly Reinforced Rectangular Beams

From

Doubly Reinforced Beams

Strength

strong

Relationship

Doubly reinforced design superimposes two singly-reinforced couples when a single layer cannot carry the moment

To

Flexural Design

From

Doubly Reinforced Beams

Strength

moderate

Relationship

Applied when standard design reaches maximum steel ratio or when deflection control is needed

To

Singly Reinforced Rectangular Beams

From

T-Beams & Flanged Sections

Strength

strong

Relationship

T-beam analysis simplifies to rectangular beam analysis when stress block lies entirely within the flange

To

Singly Reinforced Rectangular Beams

From

Material Properties & Assumptions

Strength

strong

Relationship

Material properties (f'c, fy, β₁) are inputs to all stress block and moment calculations

To

Steel Ratio Limits & Ductility

From

Material Properties & Assumptions

Strength

strong

Relationship

β₁, f'c, and fy directly define the equations for ρb, ρmax, and ρmin

To

Steel Ratio Limits & Ductility

From

Failure Modes & Behavior

Strength

strong

Relationship

Tension-controlled vs compression-controlled failure modes result from whether steel or concrete ratio dominates; defines ductility and φ

To

Flexural Design

From

Failure Modes & Behavior

Strength

moderate

Relationship

Understanding failure modes justifies why ductile (tension-controlled) design is preferred and why over-reinforcement must be avoided

To

Steel Ratio Limits & Ductility

From

NSCP 2015 & ACI 318

Strength

strong

Relationship

NSCP 2015 specifies the exact definitions of ρmin, ρmax (based on εt = 0.005) and the strength reduction factor φ

To

Flexural Design

From

NSCP 2015 & ACI 318

Strength

strong

Relationship

NSCP 2015 mandates spacing, cover, bar diameter limits, and material specifications used in design process

To

T-Beams & Flanged Sections

From

NSCP 2015 & ACI 318

Strength

moderate

Relationship

NSCP 2015 defines effective flange width, stress block location, and when T-beam analysis is required

To

Doubly Reinforced Beams

From

Singly Reinforced Rectangular Beams

Strength

strong

Relationship

When a singly reinforced beam cannot provide enough capacity, compression steel is added; analysis extends the single-steel method

To

Doubly Reinforced Beams

From

T-Beams & Flanged Sections

Strength

moderate

Relationship

Doubly reinforced T-beams combine flange compression contribution with added compression steel for very large moments

To

Failure Modes & Behavior

From

Flexural Analysis

Strength

strong

Relationship

Analysis computes εt to determine whether section is tension- or compression-controlled, thus explaining failure behavior

To

Flexural Design

From

Coefficient of Resistance Rn

Strength

strong

Relationship

Rn = Mu/(φbd²) is the key intermediate quantity that bridges load demand to required steel ratio